Matrix-based numerical modelling of financial differential equations

Differentiation matrices provide a compact and unified formulation for a variety of differential equation discretisation and time-stepping algorithms. This paper illustrates their use for solving three differential equations of finance: the classic Black-Scholes equation (linear initial-boundary value problem), an American option pricing problem (linear complementarity problem), and an optimal maintenance and shutdown model (nonlinear boundary value problem with free boundary). We present numerical results that show the advantage of an L-stable time-stepping method over the Crank-Nicolson method, and results that show how spectral collocation methods are superior for boundary value problems with smooth solutions, while finite difference methods are superior for option-pricing problems.